Suppose \(x\) is a solution to \(x^2 + 1 = 7x\) . What is the sum of \(x\) and its reciprocal?

Guest Oct 22, 2019

#1**+1 **

You can simply find x and calculate

Eventually, this equals 7.

However, there is a faster way. Can you find out yourself?

You are very welcome!

:P

CoolStuffYT Oct 22, 2019

#2**0 **

I think using the quadratic formula, we can find what x is equal to.

so

x^2+1=7x

Subtract 7x from both sides

x^2-7x+1

(Notice it is in the form ax^2+bx+c)

Apply the quadratic formula

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

You will eventually get

x=6.85410196 approx.=6.85

Now I don't understand what do you mean what is the sum of x?

Its reciprocal is just 1/6.85 i guess.

Guest Oct 23, 2019

#3**+1 **

Rearrange as

x^2 - 7x = -1

Complete the square on x

x^2 - 7x + 49/4 = -1 + 49/4

(x - 7/2)^2 = 45/4 take the positive root

x - 7/2 = √45 / 2

x - 7/2 = 3√5/2

x = [ 7 + 3√5] / 2

Its reciprocal is 2 / [ 7 + 3√5]

So

[ 7 + 3√5] / 2 + 2 / [ 7 + 3√5] =

[ 7 + 3√5] ^2 + 4

_______________ =

2 [ 7 + 3√5]

[ 49 + 42√5 + 45 + 4 ]

__________________ =

2 [ 7 + 3√5]

[ 98 + 42 √5 ]

____________ =

2 [ 7 + 3√5]

49 + 21√5

_________ =

7 + 3√5

[49 + 21√5] [ 7 - 3√5]

__________________ =

49 - 45

343 + 147√5 - 147√5 - 63*5

_______________________ =

4

343 - 315

_________ =

4

28

__ =

4

7

CPhill Oct 23, 2019